ABSTRACT

The purpose of the present paper is to sudy the pseudo-quasi-conformally flat, $\tilde{C}\cdot S=0$ , pseudo-quasi-conformal ϕ-symmetric and pseudo-quasi-conformal ϕ-recurrent 3-dimensional (LCS)_n maifolds.

Subject Areas:

Geometry

Keywords:

Lorentzian Metric, (LCS)_n-Manifolds, ϕ-Symmetric, Three-Dimensional ϕ-Recurrent, Einstein Manifold

1. Introduction

The notion of Lorentzian concircular structure manifolds (briefly (LCS)_n-manifold) was introduced by [1] , investigated the application to the general theory of relativity and cosmology with an example, which generalizes the notion of LP-Sasakian manifold introduced by Matsumoto [2] . The notion of Riemannian manifold has been weakened by many authors in a different extent [3] - [8] .

Shaikh and Jana in 2005 [9] introduced and studied a tensor field, called Pseudo-Quasi-Conformal curvature tensor $\tilde{C}$ on a Riemannian manifold of dimension ( $n\ge 3$ ). This includes the Projective, Quasi-conformal, Weyl conformal and Concircular curvature tensor as special cases. Recently Kundu and others [10] [11] studied pseudo-quasi-conformal curvature tensor on P-Sasakian manifolds.

In this paper, we consider a (LCS)_n-manifold satisfying certain conditions on the 3-dimensional pseudo-quasi-conformal curvature tensor. In section 2, we have the preliminaries. In Section 3, we studied a 3-dimensional pseudo-quasi-conformally flat (LCS)_n-manifold and proved that the manifold is η-Einstein and it is not a conformal curvature tensor. In section 4, we proved a 3-dimensional pseudo-quasi-conformal (LCS)_n-manifold satisfies $\tilde{C}\cdot S=0$ ; this reduces to η-Einstein and it is not a conformal curvature tensor. In section 5, we studied 3-dimensional pseudo-quasi-conformal ϕ-symmetric (LCS)_n-manifold with constant scalar curvature and obtained the manifold is Einstein (provided $p\ne 0$ ). In section 6, we studied a pseudo-quasi-conformal ϕ-recurrent (LCS)_n-manifold with constant scalar curvature, which generalizes the notion of ϕ-symmetric (LCS)_n-manifold.

2. Preliminaries

An n-dimensional Lorentzian manifold M is a smooth connected paracompact Hausdorff manifold with a Lorentzian metric g, that is, M admits a smooth symmetric tensor field g of type $\left(\mathrm{0,2}\right)$ such that for each point $p\in M$ , the tensor $g_p\mathrm{<mo>\; :\; </mo>}{T}_{p}M\times T_{p}M\to \Re$ is a non-degenerate inner product of signature $\left(-,+,\cdots ,+\right)$ , where $T_{p}M$ denotes the tangent vector space of M at p and R is the real number space. A non zero vector $v\in T_{p}M$ is said to be timelike (resp., non-spacelike, null, space like) if it satisfies $g_p\left(v,v\right)<0$ [12] .

Definition 2.1. In a Lorentzian manifold $\left(M\mathrm{,}g\right)$ a vector field P defined by

$g\left(X,P\right)=A\left(X\right),$

for any $X\in \chi \left(M\right)$ is said to be a concircular vector field if

$\left({\nabla }_{X}A\right)\left(Y\right)=\alpha g\left(X\mathrm{,}Y\right)+w\left(X\right)A\left(Y\right)\mathrm{,}$

where $\alpha$ is a non-zero scalar and w is a closed 1-form.

Let M be a Loretzian manifold admitting a unit timelike concircular vector field $\xi$ is called the characteristic vector field of the manifold. Then we have

$g\left(\xi ,\xi \right)=-1.$ (2.1)

Since $\xi$ is a unit concircular vector field, it follows that there exist a non-zero 1-form $\eta$ such that for

$g\left(X,\xi \right)=\eta \left(X\right),$ (2.2)

the equation of the following form holds

$\left({\nabla }_X\eta \right)\left(Y\right)=\alpha \left\{g\left(X\mathrm{,}Y\right)+\eta \left(X\right)\eta \left(Y\right)\right\}\left(\alpha \ne 0\right)\mathrm{,}$ (2.3)

for all vector field $X\mathrm{,}Y$ , where $\nabla$ denotes the operator of covariant differentiation with respect to the Lorentzian metric g and $\alpha$ is a non-zero scalar function satisfies

${\nabla }_X\alpha =\left(X\alpha \right)=d\alpha \left(X\right)=\rho \eta \left(X\right),$ (2.4)

$\rho$ being a certain scalar function given by $\rho =-\left(\xi \alpha \right)$ . Let us put

$\varphi X=\frac{1}{\alpha }{\nabla }_X\xi ,$ (2.5)

then from (2.3) and (2.5), we have

$\varphi X=X=\eta \left(X\right)\xi ,$ (2.6)

which tell us that $\varphi$ is a symmetric $\left(\mathrm{1,1}\right)$ tensor. thus the Lorentzian manifold M together with the unit timelike concircular vector field $\xi$ , its associated 1-form $\eta$ and $\left(\mathrm{1,1}\right)$ -type tensor field $\varphi$ is said to be a Lorentzian concircular structure manifold (briefly (LCS)_n-manifold) [1] . Especially, we take $\alpha =1$ , then we can obtain the LP-Sasakian structure of Matsumoto [2] . In a (LCS)_n-manifold, the following relation hold [1] .

$\eta \left(\xi \right)=-1,\varphi \xi =0,\eta \left(\varphi X\right)=0,$ (2.7)

$g\left(\varphi X,\varphi Y\right)=g\left(X,Y\right)+\eta \left(X\right)\eta \left(Y\right),$ (2.8)

$\eta \left(R\left(X\mathrm{,}Y\right)Z\right)=\left({\alpha }^2-\rho \right)\left[g\left(Y\mathrm{,}Z\right)\eta \left(X\right)-g\left(X\mathrm{,}Z\right)\eta \left(Y\right)\right]\mathrm{.}$ (2.9)

In a three dimensional (LCS)_n-manifolds, the following relation holds [13] .

$\begin{array}{c}R\left(X,Y\right)Z=g\left(Y,Z\right)QX-g\left(X,Z\right)QY+S\left(Y,Z\right)X-S\left(X,Z\right)Y\\ -\frac{r}{2}\left[g\left(Y,Z\right)X-g\left(X,Z\right)Y\right],\end{array}$ (2.10)

$S\left(X\mathrm{,}\xi \right)=2\left({\alpha }^2-\rho \right)\eta \left(X\right)\mathrm{,}$ (2.11)

$Q\xi =2\left({\alpha }^2-\rho \right)\xi ,$ (2.12)

$QX=\left[-\left({\alpha }^2-\rho \right)+\frac{r}{2}\right]X+\left[-3\left({\alpha }^2-\rho \right)+\frac{r}{2}\right]\eta \left(X\right)\xi .$ (2.13)

The pseudo-quasi-conformal curvatur tesor $\tilde{C}$ is defined by [14] .

$\begin{array}{c}\tilde{C}\left(X,Y\right)Z=\left(p+d\right)R\left(X,Y\right)Z+\left(q-\frac{d}{n-1}\right)\left[S\left(Y,Z\right)X-S\left(X,Z\right)Y\right]\\ +q\left[g\left(Y,Z\right)QX-g\left(X,Z\right)QY\right]\\ -\frac{r}{n\left(n-1\right)}\left\{p+2\left(n-1\right)q\right\}\left[g\left(Y,Z\right)X-g\left(X,Z\right)Y\right],\end{array}$ (2.14)

where $X\mathrm{,}Y\mathrm{,}Z\in \chi \left(M\right)$ , $R\mathrm{,}S\mathrm{,}Q$ and r are the curvature tensor, the Ricci tensor, the symmetric endomorphism of the tangent space at each point corresponding to the Ricci tensor S and the scalar curvature, i.e, $g\left(QX,Y\right)=S\left(X,Y\right)$ and $p\mathrm{,}q\mathrm{,}d$ are real constants such that $p^2+q^2+d^2>0$ .

In particular, if (1) $p=q=0,d=1$ ; (2) $p\ne 0,q\ne 0,d=0$ ; (3) $p=1,q=-\frac{1}{n-2},d=0$ ; (4) $p=1,q=d=0$ ; then $\tilde{C}$ reduces to the projective

curvature tensor; quasi-conformal curvature tensor; conformal curvature tensor and concircular curvature tensor, respectively.

3. 3-Dimensional Pseudo-quasi-conformally Flat (LCS)_n-Manifold

Definition 3.2. An n-dimensional ( $n\ge 3$ ) (LCS)_n-manifold M is called a pseudo-quasi-conformally flat, if the condition $\tilde{C}\left(X,Y\right)Z=0$ , for all $X\mathrm{,}Y\mathrm{,}Z\in T_{p}M.$

Let us consider the three dimensional (LCS)_n-manifold M is a pseudo-quasi-conformally flat, then from (2.6), (2.7) and (2.8) relation to (2.10) that

$\begin{array}{l}\left(p+d\right)R\left(X\mathrm{,}Y\right)Z+\left(q-\frac{d}{2}\right)\left[S\left(Y\mathrm{,}Z\right)X-S\left(X\mathrm{,}Z\right)Y\right]\\ +q\left[g\left(Y,Z\right)QX-g\left(X,Z\right)QY\right]-\frac{r}{6}\left\{p+4q\right\}\left[g\left(Y,Z\right)X-g\left(X,Z\right)Y\right]=0,\end{array}$ (3.1)

Putting $Z=\xi$ in (3.1) and by using (2.10), (2.11), we get

$\begin{array}{l}\left(p+d+q\right)\left[\eta \left(Y\right)QX-\eta \left(X\right)QY\right]\\ +\left[2\left({\alpha }^2-\rho \right)\left(p+\frac{d}{2}+q\right)-\left(2p+3d+4q\right)\frac{r}{2}\right]\eta \left(Y\right)X-\eta \left(X\right)Y=\mathrm{0,}\end{array}$ (3.2)

again plugging $Y=\xi$ in (3.2) by using (2.12) and taking inner product with respect to W, we get

$S\left(X,W\right)=\frac{1}{p+d+q}\left\{Ag\left(X,W\right)+B\eta \left(X\right)\eta \left(W\right)\right\}.$ (3.3)

where

$A=\left\{p\left[2\left({\alpha }^2-\rho \right)-\frac{r}{3}\right]+d\left[\left({\alpha }^2-\rho \right)-\frac{r}{2}\right]+2q\left[\left({\alpha }^2-\rho \right)-\frac{r}{3}\right]\right\},$

and $B=\left\{p\left[4\left({\alpha }^2-\rho \right)-r\right]+3d\left[\left({\alpha }^2-\rho \right)-\frac{r}{2}\right]+2q\left[4\left({\alpha }^2-\rho \right)-r\right]\right\}.$

Hence we can state the following theorem.

Theorem 3.1. Let M be a 3-dimensional pseudo-quasi-conformally flat (LCS)_n-manifold is an η-Einstein manifold, provided pseudo-quasi-conformal curvature tensor is not a conformal curvature tensor [15] (p = 1, q = −1 and d = 0).

4. 3-Dimensional Pseudo-quasi-conformal (LCS)_n-Manifold Satisfies $\tilde{C}\cdot S=0$

Let us consider a 3-dimensional Riemannian manifold which satisfies the condition

$\tilde{C}\left(X,Y\right)\cdot S=0.$ (4.1)

Then we have

$S\left(\tilde{C}\left(X,Y\right)U,V\right)+S\left(U,\tilde{C}\left(X,Y\right)V\right)=0.$ (4.2)

Put $X=\xi$ in (4.2) by using (2.10), (2.11), (2.12) and (2.14) and also on plugging $U=\xi$ , we get

$\begin{array}{l}4\left(p+d+q\right){\left({\alpha }^2-\rho \right)}^2\eta \left(V\right)\eta \left(Y\right)+\left(p+q+d\right)S\left(QY\mathrm{,}V\right)\\ -\frac{r}{6}\left(4p+3d+4q\right)S\left(Y\mathrm{,}V\right)-2\left({\alpha }^2-\rho \right)\left(p+d+q\right)\eta \left(V\right)\eta \left(QY\right)\\ +\frac{r}{3}\left({\alpha }^2-\rho \right)\left(4p+3d+4q\right)g\left(Y\mathrm{,}V\right)=\mathrm{0,}\end{array}$ (4.3)

by using (2.13) in (4.3), we get

$S\left(V,Y\right)=Ag\left(V,Y\right)+B\eta \left(V\right)\eta \left(Y\right).$ (4.4)

where

$A=\frac{2r\left({\alpha }^2-\rho \right)\left(4p+3d+4q\right)}{6\left({\alpha }^2-\rho \right)\left(p+d+q\right)+r\left(p+q\right)},$

and $B=\frac{6\left({\alpha }^2-\rho \right)\left[r-6\left({\alpha }^2-\rho \right)\right]\left(p+d+q\right)}{6\left({\alpha }^2-\rho \right)\left(p+d+q\right)+r\left(p+q\right)}\mathrm{.}$

Hence we can state the following theorem.

Theorem 4.2. Let a 3-dimensional pseudo-quasi-conformal (LCS)_n-manifold satisfying $\tilde{C}\cdot S\mathrm{<mo>\; =\; </mo>0}$ is an η-Einstein manifold.

5. On 3-Dimensional Pseudo-quasi-conformal ϕ-Symmetric (LCS)_n-Manifold

Definition 5.3. An (LCS)_n-manifold is said to be pseudo-quasi-conformal ϕ-symmetric if the condition

${\varphi }^2\left(\left({\nabla }_W\tilde{C}\right)\left(X,Y\right)Z\right)=0,$ (5.1)

for any vector field $X,Y,Z,W\in TpM$ .

Let us consider 3-dimensional (LCS)_n-manifold of a pseudo-quasi-conformal curvature tensor has the following from (2.6), we get

$\left({\nabla }_W\tilde{C}\right)\left(X\mathrm{,}Y\right)Z+\eta \left(\left({\nabla }_W\tilde{C}\right)\left(X\mathrm{,}Y\right)Z\right)\xi =0.$ (5.2)

which follows that

$g\left(\left({\nabla }_W\tilde{C}\right)\left(X\mathrm{,}Y\right)Z\mathrm{,}U\right)+\eta \left(\left({\nabla }_W\tilde{C}\right)\left(X\mathrm{,}Y\right)Z\right)\eta \left(U\right)=0.$ (5.3)

By virtue of (2.10), (2.11) and (2.14) and contracting we get

$\begin{array}{l}\left(q-d\right)\left({\nabla }_{W}S\right)\left(Y\mathrm{,}Z\right)+\frac{drW}{6}\left[\left(2p+2q+3d\right)g\left(Y\mathrm{,}Z\right)\left(4p+3d+4q\right)\eta \left(Y\right)\eta \left(Z\right)\right]\\ +\left(-\left(p+d\right)+q\right)\eta \left(Z\right)\left({\nabla }_{W}S\right)\left(Y\mathrm{,}\xi \right)+\left(-p-\frac{3d}{2}+q\right)\eta \left(Y\right)\left({\nabla }_{W}S\right)\left(Z\mathrm{,}\xi \right)=0.\end{array}$

(5.4)

On plugging $Z=\xi$ in (5.4), gives

$S\left(Y\mathrm{,}W\right)=2\left({\alpha }^2-\rho \right)g\left(Y\mathrm{,}W\right)-\frac{1}{p\alpha }\left(p+q\right)\eta \left(Y\right)\frac{drW}{3}\mathrm{.}$ (5.5)

If the manifold has a constant scalar curvature r, then $drW=0$ .

Hence the Equation (5.5) turns into

$S\left(Y\mathrm{,}W\right)=2\left({\alpha }^2-\rho \right)g\left(Y\mathrm{,}W\right)\mathrm{.}$ (5.6)

Hence we can state the following:

Theorem 5.3. Let M be a 3-dimensional pseudo-quasi-conformal ϕ-symmetric (LCS)_n-manifold with constant scalar curvature, then the manifold is reduces to a Einstein manifold.

6. 3-Dimensional Pseudo-quasi-conformal ϕ-Recurrent on (LCS)_n-Manifold

Definition 6.4. An (LCS)_n-manifold is said to be pseudo-quasi-conformal ϕ-recurrent if

${\varphi }^2\left(\left({\nabla }_W\tilde{C}\right)\left(X\mathrm{,}Y\right)Z\right)=A\left(W\right)\tilde{C}\left(X\mathrm{,}Y\right)Z\mathrm{,}$ (6.1)

for any vector field $X\mathrm{,}Y\mathrm{,}Z\mathrm{,}W\in TpM$ . If $A\left(W\right)=0$ then pseudo-quasi-conformal ϕ-recurrent reduces to ϕ-symmetric.

Let us consider a 3-dimensional pseudo-quasi-conformal ϕ-recurrent (LCS)_n-manifold. Then by virtue of (2.6) and (6.1), we have

$\left({\nabla }_W\tilde{C}\right)\left(X\mathrm{,}Y\right)Z+\eta \left(\left({\nabla }_W\tilde{C}\right)\left(X\mathrm{,}Y\right)Z\right)\xi =A\left(W\right)\tilde{C}\left(X\mathrm{,}Y\right)Z\mathrm{,}$ (6.2)

from which it follows that

$g\left(\left({\nabla }_W\tilde{C}\right)\left(X,Y\right)Z,U\right)+\eta \left(\left({\nabla }_W\tilde{C}\right)\left(X,Y\right)Z\right)\eta \left(U\right)=A\left(W\right)g\left(\tilde{C}\left(X,Y\right)Z,U\right).$ (6.3)

By virtue of (2.10), (2.11) and (2.14) and contracting, also plugging $Z=\xi$ , we get

$\begin{array}{l}S\left(Y,W\right)\\ =2\left({\alpha }^2-\rho \right)g\left(Y,W\right)-\frac{2}{3}\left[\frac{r\left(2p+3d-q\right)+6\left(p+q\right)\left({\alpha }^2-\rho \right)}{\left(2p+d+2q\right)\alpha }\right]\eta \left(Y\right)A\left(W\right)\\ +drW\eta \left(Y\right)\frac{2\left(p+d\right)}{\alpha \left(2p+d+2q\right)},\end{array}$ (6.4)

again putting $Y=\xi$ in (6.4), we get

$A\left(W\right)=\frac{3\left(p+d\right)}{r\left(q-2p-3d\right)-6\left(p+q\right)\left({\alpha }^2-\rho \right)}drW\mathrm{.}$ (6.5)

If the manifold has a constant scalar curvature r, then $drW=0$ . Hence the Equation (6.5) turns into

$A\left(W\right)=0.$ (6.6)

Using (6.6) in (6.1), we get

${\varphi }^2\left(\left({\nabla }_W\tilde{C}\right)\left(X\mathrm{,}Y\right)Z\right)=0.$ (6.7)

Hence we can state the following:

Theorem 6.4. If M is a 3-dimensional pseudo-quasi-conformal ϕ-recurrent (LCS)_n-manifold with constant scalar curvature, then it is ϕ-symmetric.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Murthy, B. and Venkatesha (2019) On 3-Dimensional Pseudo-Quasi-Conformal Curvature Tensor on (LCS)_n-Manifolds. Open Access Library Journal, 6: e5474. https://doi.org/10.4236/oalib.1105474

References